intersection between box and convex set

Question

I have a task to make an intersection between the convex set and the box , the equation describing the convex set is r1r2r3 - r1 -r2 - r3 + 2 cos(phi) > 0, phi = 45 ,and I plot it using p2 = ContourPlot3D[xyz - x - y - z + 2*Cos[45] == 0, {x, 0, 4}, {y, 0, 4}, {z, 0, 4}]. but , I have no idea how to plot the box and how to make the intersection between them to obtain the figure (b). note that to plot the box r1=3.7, r2=1.3 , r3=3. can anyone help me?

Answer 1

Here wee use {x, 0.5, r1}, {y, 0.5, r2}, {z, 0.5, r3} to cut the region x*y*z - x - y - z + 2*Cos[45] > 0 and set the PlotRange to some large space PlotRange->{{0, r1 + 1}, {0, r2 + 1}, {0, r3 + 1}} and then we get the second plot intersection.

r1 = 3.7; r2 = 1.3; r3 = 3;
intersection = 
 RegionPlot3D[
  x*y*z - x - y - z + 2*Cos[45] > 0, {x, 0.5, r1}, {y, 0.5, r2}, {z, 
   0.5, r3}, PlotRange -> {{0, r1 + 1}, {0, r2 + 1}, {0, r3 + 1}}]

regs = RegionPlot3D[{0 <= x <= r1 && 0 <= y <= r2 && 0 <= z <= r3, 
    x*y*z - x - y - z + 2*Cos[45] > 0}, {x, 0.5, r1 + 1}, {y, 0.5, 
    r2 + 1}, {z, 0.5, r3 + 1}, 
   PlotStyle -> {Directive[Gray, Opacity[.2]], 
     Directive[Pink, Opacity[.2]]}, Mesh -> None];
Show[regs, intersection]